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Definition

Let and be two algebras in a variety of algebras. Then, a map from to is termed a homomorphism of universal algebras if where is a member of the operator domain corresponding to the variety.

The on the left is in and the on the right is in .

Examples

Homomorphism of magmas

Consider the variety of magmas: a magma is a set equipped with a binary operation. The operator domain here consists of a single operator: the binary operation of multiplication (denoted as ). Thus, given a map → of magmas, is a homomorphism if and only if, for every in :

The on the left is in and the on the right is in .

Here, plays the role of . Note that we have used infix notation for as opposed to prefix notation for , which is why the expression looks somewhat different.

Homomorphism of monoids

Consider the variety of monoids: a monoid is a set equipped with a binary operation , as well as a constant called the neutral element, such that:

A map → of monoids is termed a homomorphism of monoids of and \phi(e) = e</math>.

Note that since every monoid is also a magma (by only looking at the binary operation) we can also talk of magma-theoretic homomorphisms of monoids. However, it is not true that any magma-theoretic homomorphism is also a homomorphism of monoids. In particular, the neutral element may not go to the neutral element.

Homomorphism of groups

A homomorphism of groups is a map from one group to another that preserves: the binary operation, the inverse operation and the neutral element. It turns out that any magma-theoretic homomorphism between groups is also a homomorphism of groups. Hence, we can also define a homomorphism of groups as a set-theoretic map between groups that preserves the binary operation.